Surfaces with parallel mean curvature vector in $\mathbb{S}^{2}×\mathbb{S}^{2}$ and $\mathbb{H}^{2}×\mathbb{H}^{2}$

Torralbo, Francisco; Urbano, Francisco J.

doi:10.1090/s0002-9947-2011-05346-8

Cited by 19 publications

(1 citation statement)

References 14 publications

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“…and surfaces with parallel mean curvature vector in S 2 × S 2 (cf. [18], etc.). In hypersurface case (i.e., codimension 1), it is remarable that Urbano [20] classified the homogenous hypersurfaces and the isoparametric hypersurfaces of S 2 × S 2 , and he also studied the hypersufaces of S 2 × S 2 with constant principal curvatures and gave some other important classification results.…”

Section: Introductionmentioning

confidence: 98%

Affine hypersurfaces with constant sectional curvature

Antić

Vrancken

et al. 2021

Pacific J. Math.

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In this paper, we classify the hypersurfaces of S 2 × S 2 with constant sectional curvature. By applying the so-called Tsinghua principle, which was first discovered by the first three authors in 2013 at Tsinghua University, we prove that the constant sectional curvature can only be 1 2 and the product angle function C defined by Urbano is identically zero. We show that any such hypersurface is a parallel hypersurface of a minimal hypersurface in S 2 × S 2 with C = 0, and we establish a one-to-one correspondence between the involving minimal hypersurface and the famous "sinh-Gordon equation"As a byproduct, we give a complete classification of the hypersurfaces of S 2 × S 2 with constant mean curvature and constant product angle function C.

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Section: Introductionmentioning

confidence: 98%